Munich Personal RePEc Archive

Selecting between different productivity

measurement approaches: An application

using EU KLEMS data

Giraleas, Dimitris and Emrouznejad, Ali and Thanassoulis,

Emmanuel

Aston Business School, Aston University

March 2012

Online at https://mpra.ub.uni-muenchen.de/37965/

MPRA Paper No. 37965, posted 11 Apr 2012 02:38 UTC

1

Selecting between different productivity

measurement approaches: An application

using EU KLEMS data

Dimitris Giraleas1, Ali Emrouznejad and Emmanuel Thanassoulis

Operations and Information Management, Aston Business School, Aston University, Birmingham, UK

Abstract: Over the years, a number of different approaches were developed

to measure productivity change, both in the micro and the macro setting.

Since each approach comes with its own set of assumptions, it is not

uncommon in practice that they produce different, and sometimes quite

divergent, productivity change estimates. This paper introduces a framework

that can be used to select between the most common productivity

measurement approaches based on a number of characteristics specific to

the application/dataset at hand; these were selected based on the results of

previous simulation analysis that examined the accuracy of different

productivity measurement approaches under different conditions. The

characteristics in question include input volatility through time, the extent of

technical inefficiency and noise present in the dataset and whether the

parametric approaches are likely to suffer from functional form miss-

specification and are examined using a number of well-established

diagnostics and indicators. Once assessed, the most appropriate approach

can be selected based on its relative accuracy under these conditions;

accuracy can in turn be assessed using simulation analysis, either previously

published or designed specifically to emulate the characteristics of the

application/dataset at hand. As an example of how this selection framework

can be implemented in practice, we assess the productivity performance of a

number of EU countries using the EU KLEMS dataset.

Keywords: Data envelopment analysis, Productivity and competitiveness,

Simulation, Stochastic Frontier Analysis, Growth accounting

1 Corresponding Author: Email address: giralead@aston.ac.uk (D Giraleas)

2

1 Introduction

Increasing productivity is of paramount importance to policy makers in both

the micro (ie company-level) and macro (ie industry- and economy-level)

settings, since productivity growth leads to increased prosperity both in the

short term and long-run. In fact, according to neoclassical theory, productivity

growth is the only sustainable source of growth in the long-run since factor

accumulation provides only decreasing returns to growth (Solow, 1957).

Given the importance of productivity growth in both the micro but especially in

the macro level, where increasing input utilisation is more difficult, it is critical

for policy makers to accurately measure its progress. Accurate measures of

productivity change are a necessary requirement for any analysis that looks

for the likely drivers of productivity growth; once these drivers are identified,

policy makers can work on fostering the development of such drivers in the

economy, leading to increased economic prosperity.

There are a number of approaches that could be used to measure productivity

change, both in the macro (ie economy- or industry-wide) and micro (eg

company or department) level, and each has its own strengths and

weaknesses (Knox Lovell, 1996). Usually, the analysis will adopt more than

one of these approaches, so that comparisons between the different

productivity change estimates are possible (see for example (Odeck, 2007)).

It is not uncommon in such applications that the different approaches result in

different estimates; in some cases, these differences can be quite substantial

(Coelli, 2002). In those instances, the analysis would need to be able to put

forward an informed view on why the various approaches come up with such

divergent views and, more importantly, which estimates are likely to be more

accurate. The aim of this paper is to suggest a framework which could be

used to select between the competing estimates, based on the likely accuracy

of the adopted approaches when taking into account the

characteristics/conditions specific to the application at hand. Although such a

framework is also applicable in the micro setting, the focus for this paper is on

assessing the relative accuracy of the different productivity estimates in the

economy-wide setting.

3

To develop this framework, we heavily rely on the simulation experiments

undertaken in Giraleas et al.2 (forthcoming) (referred to from now on as GET),

which examined the overall accuracy of some of the most commonly-used

productivity measurement approaches under a number of different conditions.

The experiments revealed that the relative accuracy of each approach heavily

depends on said conditions; therefore, if we could identify which

characteristics/conditions are prevalent in the application/dataset currently

examined, we could use the simulation evidence to select the productivity

estimates that are likely to be more accurate. This paper proposes a set of

readily available diagnostic tests and/or indicators that could be used to asses

the aforementioned conditions/characteristics; when combined, these

diagnostic tests and indicators would constitute the selection framework. The

application of this framework is demonstrated using the EU KLEMS dataset

(EU KLEMS, 2008).

The paper is structured as follows. Section 2 presents the selection

framework and how it can be applied in practice. Section 3 assesses the

annual productivity change for a group of countries by a number of

approaches, using the EU KLEMS dataset, and compares the resulting

estimates. Section 4 demonstrates an application of the selection framework

using the analysis in section 3 as an example. Lastly, section 5 provides a

summary of the selection framework and concludes.

2 Selection framework in the macro setting

Productivity change in the macro setting (aggregate productivity) is usually

measured utilising a single measure of aggregate output (usually expressed in

value added terms) and a limited number of aggregate inputs (labour and

capital if output is measured in value added terms). The most common

approach of measuring aggregate productivity change is growth accounting

(GA). Probably the largest contributor to the wide adoption of GA amongst

policy makers is that GA estimates can be (relatively) easily produced using

country- or sector-specific National Accounts data, without recourse to

information from outside the country or the sector examined; on the other

hand, GA requires the adoption of a number of, potentially unrealistic

2 Available at: http://mpra.ub.uni-muenchen.de/37429/

4

assumptions, most notably those relying on the existence of perfect

competition, which could lead to unreliable estimates (for a more detailed

discussion see section 3.2).

Frontier-based approaches can also be used to estimate productivity change

and unlike GA, they do not rely on the (restrictive) assumptions of perfectly

competitive markets nor do they require information on prices. On the other

hand, frontier-based approaches require information on comparators, so they

are, in a sense, more data intensive than GA. The most common frontier-

based approaches for measuring productivity change are Data Envelopment

Analysis (DEA), Corrected Ordinary Least Squares (COLS) and Stochastic

Frontier analysis; these are also the focus of this paper and are discussed in

more detail in section 3.2.

In terms of the overall accuracy of each approach, the simulation analysis

undertaken in GET demonstrated that no single approach has an absolute

advantage over another; rather, their relative accuracy depends on the

characteristics of the data generating process (DGP), or, in other words, the

characteristics of the application/dataset at hand.

2.1 Characteristics of interest

The simulation analysis undertaken in GET revealed that the most influential

characteristics of the DGP to the accuracy of the examined approaches are:

– the extend of volatility in inputs from one year to the next. Increased

volatility adversely affects the accuracy of all approaches, but DEA-based

estimates are the least affected, while the GA estimates are the most

affected;

– the extend of inefficiency present in the sample. Increased levels of

technical inefficiency have only a small negative effect on the accuracy of

COLS- and DEA-derived productivity estimates, but a larger impact on

GA and SFA-based estimates (for the SFA estimates the change in

accuracy is co-dependent on the extend of measurement error-noise in

the data);

– the extend of measurement error/noise in the data. Increased levels of

noise have detrimental effect on the accuracy of all approaches, but the

overall effect depends on the extend of technical inefficiency also present

5

and on whether the parametric approaches are likely to suffer from

functional form miss-specification;

– whether the parametric approaches are likely to suffer from functional

form miss-specification; Functional form misspecification has a severe

negative impact on the accuracy of all parametric approaches.

The first step of the proposed selection framework is to assess how prevalent

the above characteristics are (if at all) in the application/dataset at hand. This

assessment is not straightforward; in fact, it is impossible to determine with

certainty at least some of the characteristics in question, given that all

productivity measurement approaches examined rely on certain implicit or

explicit assumptions, which are made prior to the actual analysis, that directly

influence the estimates used to assess said characteristics. For example, both

COLS and DEA are so-called ‘deterministic approaches’, meaning that they

assume that there is no measurement error/noise in the sample data. Another

example is that most frontier-based approaches automatically assume that

there is some inefficiency (SFA is the exception, as it can test for the

presence inefficiency), while GA assumes that there is no inefficiency in the

sample data.

Despite the above concerns, there are a number of simple diagnostic

tests/indicators that can provide useful information on the presence or

prevalence of the characteristics in question.

Input volatility

This is relatively easy to assess by simply examining the summary statistics

(namely the average and the standard deviation) of the annual change in

inputs of each assessed unit.

Technical inefficiency

The various frontier-based approaches can readily provide estimates of

technical inefficiency as well as estimates of productivity change; we can

make use of these estimates to assess the possible extent of technical

inefficiency in the dataset/application at hand. Since we are not certain which

approach is likely to provide the most accurate efficiency estimates, we

assess this characteristic by simply averaging the different efficiency

estimates derived from all adopted approaches.

6

In general, it is difficult to assess the extent of technical inefficiency with a

high degree of accuracy, since the performance measurement approaches

examined measure it residually. As such, the way they construct the efficiency

frontier (or the production possibility set) will always have a significant impact

on the final performance measure (efficiency or productivity estimate).

Nevertheless, although perfect accuracy is out of reach, there is a large and

growing pool of evidence in the literature that suggests that technical

inefficiency is present in the economy (for examples, see (Fried, Lovell, &

Schmidt, 2008) and (del Gatto, di Liberto, & Petraglia, 2008)) and that frontier-

based approaches can, in most cases, measure such inefficiency with a

degree of accuracy sufficient for the purposes of the selection framework (as

demonstrated by numerous simulation studies, such as (Banker, Chang, &

Cooper, 2004) and (Resti, 2000)).

Noise levels

From the approaches examined, only SFA incorporates a so-called stochastic

element in the estimation process which is able to capture the impact of

measurement error/statistical noise; as such, overall noise levels in the DGP

can be measured by the estimated standard deviation of the noise

component, denoted as σv, which can easily be extracted from the SFA

models3.

The issue with relying on this estimate to assess the extent of noise in the

dataset/application at hand is that although σv is unbiased, it is also

inconsistent in the pooled setting (because it is independent of i, ie the

observation whose technical efficiency is to be estimated; for additional

discussion see (Kumbhakar & Lovell, 2000)). It is not clear whether this issue

would materially affect the accuracy of the estimator, at least for the purposes

of this selection framework; to explore this further, we undertake a simulation

analysis to examine how reliable are the estimates of σv under certain

conditions (see section 4.3). The simulation analysis revealed that the

estimated σv is reasonably accurate under conditions similar to those

observed in the EU KLEMS dataset and can thus be used as an indicator of

the overall noise levels in this particular application.

3 The noise component/variable is assumed to be normally distributed with a mean value of zero, by construction.

7

Functional form miss-specification

This issue relates only to the parametric frontier-based approaches and can

be assessed using RESET (Ramsey Regression Equation Specification Error

Test) and by examining the statistical significance of the input coefficients.

RESET is one of the most widely-used tests to detect the presence of

functional form miss-specification. The test examines whether the inclusion of

non-linear combinations4 of either the fitted values of the regression model or

the model’s explanatory variables are statistically significant when included in

the original regression model. If they are, then the regression model is likely to

suffer from some form of misspecification. RESET is quite powerful and can

offer compelling evidence, but can only be applied when the regression model

is estimated using OLS (ordinary least squares); as such, this test can only be

used directly to assess the COLS models. However, since the input

coefficients from the SFA models are consistent estimates of the respective

OLS input coefficients, the findings of RESET as applied in the OLS

regression model also apply for the SFA model. In fact, it is not uncommon in

SFA studies to first estimate the equivalent OLS models solely for the purpose

of applying RESET to test for functional form misspecification (Jacobs, 2001).

Examining the statistical significance of the input coefficients offers a more

qualitative assessment on the possible existence of functional form miss-

specification; the intuition behind it is that if some of the input coefficients are

found to be statistically insignificant, the adopted functional form does not

match exactly to the underlying data generation process and as such the

parametric model in question could be miss-specified. It should be mentioned

that there could be a number of reasons why a variable could be assessed as

being statistically insignificant even though it is in fact part of the DGP; these

include extensive measurement error in the data or multi-collinearity amongst

the various explanatory variables. Therefore, statistically insignificant

variables in this context do not necessarily imply that the model is miss-

specified; they are however an indicator that the current model might suffer

from a number of possible shortcomings, including functional form miss-

specification, which could affect the accuracy of the derived productivity

estimates. However, as we demonstrate in section 4.4, the results of the

4 For the EU KLEMS application, we adopted the standard quadratic combinations.

8

simulation analysis indicate that when the parametric models are miss-

specified, it is also common that some input coefficients are assessed as

statistically insignificant.

2.2 Selecting between approaches

After assessing the prevalence of the above characteristics in the current

dataset/application, the next and final step of the selection framework is to

determine which of the assessed approaches offers the more accurate

productivity change under these specific conditions. This can be achieved in

two ways: we could either rely on the findings of previous simulation studies

that specifically assess the overall accuracy of different approaches under

these conditions (such as GET), or we could undertake an original simulation

analysis that uses a DGP specifically tailored to the application currently

considered. The advantage of relying on already existing studies is simplicity

and ease of implementation; however, this might come at the cost of

accuracy, in the event that the DGPs adopted by the existing studies do not

closely match the characteristics of the current application. This is avoided if

an original simulation study is undertaken, but this adds to the analytical

burden of the productivity performance assessment.

It should be mentioned that the DGP of the simulation analysis will not be able

to capture all of the peculiarities of the current application. We should of

course try to construct it in such a way as to be as similar as possible with the

current application; the simulations DGP should include the same number of

inputs and outputs, similar number of available observations (units and time

periods) and similar volatility, noise and inefficiency characteristics as the

current dataset. In addition, if we find that the parametric approaches show

evidence of functional form miss-specification even when flexible functional

forms are adopted, we should use non-smooth functional forms (such as

piecewise-linear functions5) for the simulation DGP to ensure that the

parametric approaches in the simulation analysis also suffer from functional

form miss-specification. Nevertheless, there will always be some degree of

uncertainty, since we cannot have full knowledge of the underlying DGP of the

current application (if we had, we wouldn’t need to estimate it!). For example,

5 When creating these functions, we also need to consider whether to impose the various restrictions suggested by

theory, such as monotonicity and concavity.

9

we might have relative accurate estimates of the mean and standard deviation

of technical inefficiency of the assessed units, but we cannot derive its actual

distribution; similarly, if we cannot fit a smooth function of the current dataset,

the non-smooth function adopted for the simulation analysis will not

necessarily be representative of the true underlying DGP of the current

application.

It should be stressed however that these gaps in our understanding of the

underlying DGP of our application are only an issue if they negatively affect

our ability to draw useful conclusions from the simulation analysis, be it either

original or drawn from previously published studies. In other words, these

characteristics are only important in so much as they affect the accuracy of

the resulting productivity estimates. According to the findings of GET, neither

of two examples given above were found to have a material effect in the

relative accuracy of the approaches examined; the SFA-based estimates

were very similar regardless of the distributional assumptions made by the

models, while the parametric models displayed similar loss in accuracy under

a number of different piecewise-linear DGPs.

That is not to say that the four characteristics included in our proposed

framework are the only characteristics that are likely to significantly affect the

relative accuracy of the productivity estimates. In fact, issues such as latent

heterogeneity in the assessed units (which could manifest as

heteroskedasticity in the parametric models) and variable returns to scale

could also be significant. Unfortunately, we currently do not know how these

factors affect the relative accuracy of the various approaches; as such, we

leave the assessment of those characteristics for future research.

3 Productivity change in the EU KLEMS dataset

3.1 Data

The analysis carried out in this chapter uses the dataset constructed by the

EU KLEMS (EU KLEMS 2008) project. The EU KLEMS project aims to

provide a harmonised set of indicators for the measurement and comparison

of productivity performance for a large number of, mostly, EU countries. The

dataset provides information, among others, on economy-wide level

aggregates from 1970 to 20076, based on information from each country’s

national accounts but adjusted for comparability across time and countries.

The dataset also includes GA-based total factor productivity (TFP) growth

estimates derived from the primary data, which are used in this study as

comparators to the frontier-based estimates.

This application focuses on assessing productivity at the economy-wide level.

As such, the output of choice is gross value added (GVA), which is defined as

nconsumptio teIntermediaoutput totalGrossAdded Value Gross Eq. (1)

EU KLEMS provides both economy-wide nominal GVA as well as its price

index, which can be used to calculate real GVA. Since this analysis relies on

international comparisons, real GVA is further adjusted to account for

differences in purchasing power parities (PPP) in order to enable direct

comparisons between the different countries (for more information on PPPs

see (Eurostat & OECD, 2007)). The relevant PPPs are output-specific (in this

case, calculated on the basis of GVA) and are also sourced from the EU

KLEMS dataset.

In terms of inputs, the analysis adopts ‘hours worked’ as a measure of labour

input and capital stock as a measure of capital input. The labour measure is

adjusted to take into account the differences in labour skill, which is proxied

by educational attainment. These adjustments were carried out originally by

EU KLEMS and are adopted in this instance in order to ensure comparability

between the GA estimates sourced from EU KLEMS and the frontier-based

estimates that are calculated for this analysis.

Information on capital stock is also sourced from EU KLEMS. To ensure

comparability between the countries in the sample, EU KLEMS used

harmonised depreciation rates and applied consistent capital accounting

procedures to deal with issues such as weighting between various asset

categories and rental rates. Furthermore, since the frontier-based approaches

require that capital stock is expressed in the same unit of measurement for all

countries involved, the capital stock measure is PPP-adjusted, using a capital

stock-specific PPP index (this is also included in the EU KLEMS database).

10

6 For some countries the start and end data of the period for which data is available differs, resulting in an

unbalanced panel.

11

The countries that are included in the analysis together with the time periods

for which data is available and the average values of the inputs and output are

given in table 3.1 below. Overall, the productivity growth estimates are

produced for 14 different countries, over a number of years starting from 1970

and ending in 2007; on the whole the analysis includes 375 observations

(each country in each time period as a different observation).

Table 3.1: Descriptive statistics of the dataset

Country Short code Observations

Start date

End date

PPP-adjusted Value added

(average)

Adjusted Hours worked (average)

Capital stock (average)

Australia AUS 26 1982 2007 334,732 14,996 1,012,975

Austria AUT 28 1980 2007 135,593 6,411 600,872 Czech Republic CZE 13 1995 2007 115,130 10,278 215,538

Denmark DNK 28 1980 2007 98,943 4,052 401,680

Spain ESP 28 1980 2007 533,431 24,859 2,046,906

Finland FIN 38 1970 2007 79,438 3,754 258,258

Germany GER 17 1991 2007 1,660,380 57,623 5,705,057

Italy ITA 38 1970 2007 836,748 39,704 3,221,461

Japan JPN 34 1973 2006 1,831,401 119,325 9,767,948

Netherlands NLD 29 1979 2007 282,496 10,205 987,615

Slovenia SVN 12 1995 2006 20,850 1,712 29,957

Sweden SWE 15 1993 2007 180,068 6,996 355,325 United Kingdom UK 38 1970 2007 827,492 45,309 1,890,611 United States of America USA 31 1977 2007 6,867,596 233,426 18,108,226

3.2 Methodology

Productivity change is assessed in this study using the following approaches:

– GA (estimates are provided by the EU KLEMS project),

– DEA-based circular Malmquist indices,

– COLS-based Malmquist indices, and

– SFA-based Malmquist indices.

All frontier-based approaches examined in this analysis rely on the notion of

what has come to be known as the Malmquist productivity index (Diewert,

1992), which has been used extensively in both the parametric (Kumbhakar &

Lovell, 2000) and the non-parametric (Thanassoulis, 2001) setting.

Furthermore, the productivity index produced by GA can be considered as a

special case of the Malmquist productivity index (OECD, 2001).

Growth Accounting

Growth Accounting (GA) is an index number-based approach that relies on

the neoclassical production framework, and seeks to estimate the rate of

productivity change residually, ie by examining how much of an observed rate

of change of a unit’s output can be explained by the rate of change of the

combined inputs used in the production process. There are many

modifications that could be applied to the more general GA setting ((Balk,

2008); (del Gatto, et al., 2008)); however, most applications, including the EU

KLEMS project, utilise ‘traditional’ growth accounting methods, as detailed in

OECD (OECD, 2001) and briefly described here.

GA postulates the existence of a production technology that can be

represented parametrically by a production function relating Value Added

(YGVA), to primary inputs labour (L) and capital services (K) and productivity

change (TFP), which is Hicks-neutral, such that:

TFPLKFYGVA ),( Eq. (2)

To parameterise (2), the analysis needs to adopt a number of assumptions,

such as a constant returns to scale Cobb-Douglas production function and

perfectly competitive markets; these are discussed in more detail in Annex 3

of the OECD manual (OECD, 2001).

If these assumptions hold, once the production function is differentiated with

respect to time, the rate of change in output is equal to the sum of the

weighted average of the change in inputs and the change in productivity. The

input weights are the output elasticities of each factor of production, which are

derived as the share of each input to the total value of production. Therefore,

productivity change is estimated by:

dt

KdS

dt

LdS

dt

Yd

dt

TFPd iK

iiLi

GA

ii

lnlnlnln Eq. (3)

where is the average share of labour in periods t and t-1, is the average

share of capital in t and t-1 given by

LiS

LiS

7:

12

7 EU KLEMS estimates the shares of primary inputs using arithmetic averages; alternatively, geometric averages can

also be used.

211

11

itit

it

L

it

itit

it

L

itL

Yp

Lw

Yp

LwS i Eq.(4)

211

1

,

1

,

itit

it

GAK

it

itit

it

GAK

itK

Yp

Kw

Yp

KwS i Eq.(5)

It should be noted that the price of capital is not observable; as such, EU

KLEMS, like the majority of GA applications (OECD, 2001), uses the so called

endogenous ‘user cost of capital’ to estimate the final price of capital8.

DEA-based Circular Malmquist index

The most common non-parametric approach for productivity measurement

utilises Data Envelopment Analysis (DEA) to construct Malmqusit indices (MI)

of productivity change. This approach was first proposed by Caves et al.

(Caves, Christensen, & Diewert, 1982) and later refined by Färe et al. (Färe,

Grosskopf, Norris, & Zhang, 1994).

This application utilises a circular Malmquist-type index (thereafter referred to

as circular MI), which was first proposed by Pastor et al. (Pastor & Lovell,

2005) and refined by Portela et al. (Portela & Thanassoulis, 2010).

Whereas the ‘traditional’ MI uses two reference frontiers (based on the start

and end period of the analysis) to compute the average distance between two

points, the circular MI measures this distance using a single, common frontier

as reference, which is constructed in such a way as to envelope all data

points from all periods. This common frontier is defined as the ‘meta-frontier’

and since it allows for the full envelopment of the data across, it allows for the

creation of a Malmquist-type index which is circular. Distances are measured

by standard DEA models; for this application, we employ single output (PPP-

adjusted real GVA), two input (Labour and Capital stock) constant returns to

scale models.

The main advantages of the circular MI relative to the ‘traditional’ (Färe 1994)

MI are the ease of computation and the ability to accommodate unbalanced

panel data. For a more detailed discussion, see Portela et al. (Portela &

Thanassoulis, 2010).

13

8 The endogenous user cost of capital is calculated residually, by setting capital compensation (ie the cost of capital)

to be equal to Value Added minus the labour compensation (ie the cost of labour). For more information, see the OECD manual on Measuring Capital (OECD, 2009).

Corrected OLS

Corrected OLS (COLS) is a deterministic, parametric approach and one of the

numerous ways that have been suggested to ‘correct’ the inconsistency of the

OLS-derived constant term of the regression when technical inefficiency is

present in the production process.

Two different COLS model specifications are used for this application. Both

are based on a pooled regression model (ie all observations are included in

the same model with no unit-specific effect). The first model assumes a Cobb-

Douglas functional form and is given by:

itititit tLY *lnlnln *** Eq.(6)

where it* are the estimated OLS residuals

The second COLS model specification assumes a translog functional form

and is given by:

*

222

lnlnlnln

2

1ln

2

1ln

2

1lnlnln

ititLtitKtititKL

ttitKKitLLtitKitLiit

tLtKLK

tKLtKLaY

Eq.(7)

Inefficiency estimates are derived by:

Eq.(8) )max( ***

itititu

Productivity change is calculated by adding the different components of the

Malmquist productivity index (see section Kumbhakar et al. (Kumbhakar &

Lovell, 2000)) :

dtSECddtTCddtECddtTFPdCOLS

it

COLS

it

COLS

it

COLS

it /ln/ln/ln/ln Eq.(9)

where is the COLS-estimated efficiency change, is the COLS-

estimated technical change and is the COLS-estimated scale

efficiency change.

COLS

itECCOLS

itTC

COLS

itSEC

Stochastic frontier analysis

The pre-eminent parametric frontier-based approach is Stochastic Frontier

Analysis (SFA), which was developed independently by Aigner et al. (Aigner,

Lovell, & Schmidt, 1977) and by Meeusen et al. (Meeusen & van den Broeck,

1977). The approach relies on the notion that the observed deviation from the

14

frontier could be due to both genuine inefficiency but also random effects,

including measurement error. SFA attempts to disentangle those random

effects by decomposing the residual of the parametric formulation of the

production process into noise (random error) and inefficiency.

As is the case with the COLS approach, two separate SFA model

specifications are used in this application: one that adopts a Cobb-Douglas

functional form and a second that adopts the translog. The models are very

similar to those used under COLS; the only difference lies in the specification

of the residual.

In more detail, the Cobb-Douglas model is given by:

ititititit uvtLY *** lnlnln Eq.(10)

whereas the translog model is given by:

itititLtitKtititKL

ttitKKitLLtitKitLiit

uvtLtKLK

tKLtKLaY

lnlnlnln

2

1ln

2

1ln

2

1lnlnln 222

Eq.(11)

where represents the inefficiency component (and as such ) and

represents measurement error ( ). The inefficiency component is

estimated based on the JMLS (Jondrow, Knox Lovell, Materov, & Schmidt,

1982) estimator.

itu 0itu itv

),0(~ 2

vit Nv

Two different distributions for the inefficiency component are tested:

– the exponential distribution, ) (~ uit Expu

– the half-normal distribution, ) ,0(~ 2

uit Nu

Productivity change is measured in exactly the same way as with COLS.

3.3 Results

Table 3.2 presents a summary of the annual TFP change estimates by

approach..

15

16

Table 3.2: Annual TFP change estimates

TFP measure DEA COLS

COLS translog

SFA (half-normal)

SFA (exponential)

SFA translog (half-normal)

SFA translog (exponential) GA

Mean 0.52% 0.67% 0.86% 0.82% 0.82% 0.77% 0.88% 0.54%

Std. Dev. 1.66% 1.44% 1.70% 1.14% 0.99% 1.69% 1.15% 1.44%

Note: The parametric models that are not labelled as translog adopt the Cobb-Douglas functional form.

Table 3.2 shows that for the full period, TFP has been growing at an average

annual rate of between 0.52% and 0.88%. The lowest average growth comes

from the DEA-based circular Malmquist index, while the highest estimate

comes from the translog SFA model that assumes an exponential distribution

of inefficiency.

Table 3.3: Correlation coefficients for annual TFP change

Approach Correlation measure DEA COLS

COLS translog

SFA (half-normal)

SFA (exponential)

SFA translog (half-normal)

SFA translog (exponential)

Pearson's 89%

COLS Spearman's 88%

Pearson's 88% 81% COLS translog Spearman's 90% 84%

Pearson's 84% 96% 77% SFA (half-normal) Spearman's 85% 98% 81%

Pearson's 80% 91% 72% 98% SFA (exponential) Spearman's 83% 96% 78% 99%

Pearson's 86% 80% 99% 76% 72% SFA translog (half-normal) Spearman's 90% 85% 99% 82% 80%

Pearson's 82% 76% 92% 78% 79% 94% SFA translog (exponential) Spearman's 86% 79% 93% 81% 80% 95%

Pearson's 80% 95% 79% 92% 88% 79% 75%

GA Spearman's 80% 94% 83% 93% 91% 84% 78%

The overall similarity of the average TFP change estimates between the

examined approaches is also apparent in the correlations between the

estimates, presented in table 3.3 above. GA estimates are more highly

correlated with the Cobb-Douglass COLS and SFA (half-normal) estimates

and less highly correlated with the DEA and translog-specified parametric

approaches. DEA estimates are more highly correlated with the COLS

estimates and the translog SFA estimates (but less so). It is interesting to note

17

that the translog SFA estimates are more highly correlated with each other

and their translog COLS counterparts, while they display the smallest

correlation coefficients with the estimates from the Cobb-Douglas parametric

models (COLS and SFA); this indicates that the selection of functional form to

parameterise the models can have a large effect on the TFP growth

estimates, which was also observed in the simulation analysis undertaken in

GET.

The results so far suggest that there appears to be a broad consensus

between the various approaches. However average TFP change estimates

across all countries masks the underlying variation observed at the

(individual) country level.

Table 3.4: Average annual TFP estimates by country

Country DEA COLS COLS translog

SFA (half-normal)

SFA (exponential)

SFA translog (half-normal)

SFA translog (exponential) GA

Difference between smallest and largest estimate

SVN -0.1% 1.0% -2.6% 1.2% 1.1% -3.0% -1.6% 0.9% 4.2%

CZE 0.9% 1.4% 0.3% 1.5% 1.5% 0.0% 0.1% 0.6% 1.5%

GER 1.3% 0.7% 1.8% 0.9% 0.9% 1.6% 1.2% 0.7% 1.1%

UK -0.4% 0.4% 0.2% 0.6% 0.7% 0.1% 0.5% 0.4% 1.1%

FIN 0.1% 0.7% 0.8% 0.9% 0.9% 0.9% 1.1% 1.0% 1.0%

JPN 0.5% 0.6% 1.5% 0.8% 0.8% 1.3% 1.2% 0.8% 1.0%

DNK 1.0% 0.8% 1.1% 0.9% 0.9% 1.1% 1.2% 0.3% 0.9%

NLD 0.8% 0.6% 1.0% 0.8% 0.9% 0.9% 1.3% 0.4% 0.9%

SWE 0.3% 1.1% 1.2% 1.0% 1.0% 1.0% 0.9% 0.8% 0.9%

USA 0.6% 0.5% 1.0% 0.9% 0.9% 0.8% 0.9% 0.2% 0.8%

AUT 1.2% 0.9% 1.5% 1.0% 1.0% 1.4% 1.4% 1.0% 0.6%

ESP 0.3% 0.3% 0.6% 0.4% 0.4% 0.6% 0.6% 0.0% 0.6%

ITA 0.6% 0.6% 1.0% 0.7% 0.6% 1.0% 0.9% 0.4% 0.6%

AUS 0.8% 0.7% 1.0% 0.8% 0.8% 0.9% 0.9% 0.5% 0.5%

Table 3.4 reveals that the various TFP change estimates at country level

appear to be quite different, for some countries at least. This is despite the

fact that correlations of the different estimates are still relatively high when

comparing TFP growth estimates within an individual country9. On average,

the difference between the smallest and the largest estimate is approximately

9 Not reported here, but available upon request.

18

1.1 percentage points and for some countries this can be quite larger (eg the

spread is 4.2 pc for SVN and 1.5 pc for CZE).

These, sometimes pronounced, differences in the TFP growth estimates

between the different approaches can be problematic, if such analysis were to

be used to inform policy. It is quite likely that a policy maker, upon presented

such results would enquire as to why do the various estimates differ and,

more importantly, which estimate is likely to be more accurate. The framework

described in the next section aims to facilitate this selection process.

4 Applying the selection framework to the EU KLEMS dataset

4.1 Assessing input volatility

Input volatility is the easiest characteristic to assess; it is achieved by simply

examining the annual change in inputs by country. Average input growth and

its standard deviation is summarised in the table below.

Table 4.1: Average annual growth in inputs

Country Average Growth in Labour

Standard deviation of Labour growth

Average Growth in Capital

Standard deviation of Capital growth

AUS 2.29% 1.75% 3.81% 1.44%

AUT 0.71% 1.24% 2.37% 0.29%

CZE 0.32% 1.73% 2.84% 0.33%

DNK 0.72% 1.57% 1.55% 0.92%

ESP 2.19% 2.59% 3.44% 0.87%

FIN 0.84% 2.17% 3.94% 2.26%

GER -0.35% 1.21% 2.54% 0.64%

ITA 1.04% 1.02% 2.74% 1.13%

JPN 0.64% 1.28% 4.66% 1.93%

NLD 1.42% 1.38% 2.35% 0.43%

SVN 0.89% 2.18% 6.17% 0.76%

SWE 1.15% 1.39% 3.23% 0.49%

UK 0.64% 2.20% 3.14% 0.79%

USA 1.70% 1.61% 3.29% 0.62%

Table 4.1 demonstrates that almost all countries (GER is the only exception)

have been increasing the quantities of labour inputs used in the production of

aggregate output, although the rate of increase is relatively modest. The

19

relative volatility of labour input growth, measured as the ratio of standard

deviation to average, is approximately 2.1 on average, while labour growth

volatility in absolute terms, measured only by examining the standard

deviation of the growth measure, is relatively small, averaging in

approximately 1.7%.

Most countries have also been increasing their capital stock over the period of

the analysis, with an average growth in capital inputs of 3.3%. Both relative

and absolute volatility in capital input growth is quite low (compared with

labour inputs), averaging at 0.3 and 0.9% respectively.

4.2 Assessing the extent of technical inefficiency

In order to provide an indication of how widespread technical inefficiency is in

the countries in EU KLEMS dataset, this analysis examines the various

estimates for the different approaches (and models) adopted; these are

summarised in the following table.

Table 4.2: Average technical efficiency estimates, by approach

Approach Number of observations Average

Standard deviation Minimum Maximum

DEA meta-frontier CRS 1 375 73.1% 13.8% 41.9% 100.0%

DEA meta-frontier VRS (output oriented)

1 375 79.7% 14.6% 49.5% 100.0%

DEA CRS 375 83.7% 13.2% 52.7% 100.0%

DEA VRS (output oriented) 375 89.6% 13.6% 52.7% 100.0%

COLS (Cobb-Douglas) 375 72.5% 11.9% 47.3% 100.0%

COLS (translog) 375 72.3% 10.8% 46.5% 100.0%

SFA (Cobb-Douglas, half-normal) 375 82.9% 10.8% 54.7% 96.8%

SFA (Cobb-Douglas, exponential) 375 86.6% 10.5% 56.1% 97.3%

SFA (translog, half-normal) 375 81.9% 12.1% 48.5% 100.0%

SFA (translog, exponential) 375 88.0% 10.2% 53.5% 97.4%

Note: 1 DEA meta-frontier efficiency estimates do not take into account the time dimension (technical

change and scale efficiency change) and as such are likely to be biased (downward if we assume positive technical change). They are presented here for completeness.

Direct tests for the existence of technical inefficiency are only possible for the

SFA models; with regards to this application, these tests resulted in the

rejection of the null hypothesis of no technical inefficiency in all four SFA

specifications examined.

20

Table 4.2 reveals a relative small spread of average efficiency in all the

approaches examined. The two COLS specifications display the smallest

average efficiency (approximately 72%), while the DEA output oriented VRS

models display the largest average efficiency scores (approximately 90%).

Average efficiency across all models is estimated at approximately 81% or

82% if the DEA meta-frontier efficiency scores are excluded (see note to table

4.2).

4.3 Assessing the extent of noise in the data

The relevant estimates of σv, the estimated standard deviation of the noise

component, from all the SFA models adopted for this application are provided

in the table below.

Table 4.3: Summary statistics of the σv estimate from the SFA models

SFA model Estimate of σν

Standard deviation of the σν

estimate Minimum Maximum

Cobb-Douglas, half-normal 0.075 0.010 0.058 0.098

Cobb-Douglas, exponential 0.086 0.007 0.073 0.101

Translog, half-normal 0.000 0.000 0.000 0.000

Translog, exponential 0.074 0.006 0.063 0.087

The two Cobb-Douglas models and the translog model that assumes

technical inefficiency is exponentially distributed find that the standard

deviation of the normally-distributed error term is between 0.05 to 0.1. On the

other hand, the translog SFA model that assumes half-normally distributed

technical inefficiency finds that the amount of noise in the current dataset is

negligible (σv is approximately equal to zero). This last finding appears quite

improbable; while it is true that EU KLEMS collated the various country data in

such a way as to ensure the greatest possible compatibility between the

different countries, the underlying data are still based on National Accounts

information. Since the process of data collation and aggregation required to

draw-up the National Accounts rests on a number of assumptions and

imputations10, it is expected that the data would almost always incorporate

some degree of inaccuracy11. As such, it is unlikely that the EU KLEMS

dataset is completely free of measurement error and/or statistical noise.

Since the estimate of σv is inconsistent in the pooled setting, in order to

provide some clarity on whether the use of the σv estimate is valid in this

instance, it would be helpful to observe the behaviour of the estimate under

controlled conditions through the use of simulation analysis.

This analysis utilises the same simulation framework12 presented in GET. The

simulation experiment carried out in this instance utilises a DGP constructed

in such a way that it displays similar characteristics as those observed in the

EU KLEMS dataset. In more detail, the DGP:

– is a piece-wise linear production function, since the analysis in section 4.5

below suggests that the underlying production function in the current

dataset is neither Cobb-Douglas nor translog13;

– utilises input and price data that were constructed so that they are

consistent with the level of input volatility observed in the EU KLEMS

dataset (section 4.1). In summary, input quantities and price are randomly

generated for the first period and then scaled by a random factor that

follows N~(0.0.1);

– includes a technical inefficiency component, )7/1( , which results

in average technical efficiency levels in the simulations of appr. 88%. This

is consistent with the estimates of technical inefficiency observed in the

EU KLEMS dataset, as detailed in section 4.2 of this chapter;

~ Expuit

– and lastly, includes a noise component that is randomly generated

following N~(0, 0.05), consistent with the estimates presented in table 4.3;

The summary findings of the simulation analysis are given below:

21

10

See for example the requirement to incorporate imputed rents for owners/occupiers and the methodology used to

estimate GVA from privately held corporations and unincorporated enterprises, as detailed in the ESA 1995 framework for National accounts. 11

This is also evident from the number of times that National Account information is updated, sometimes quite a few

years after the original estimates were first published. 12

As a reminder, the simulation framework in question uses 100 observations (20 DMU observed over a 5 periods)

and summarises the findings of 100 experiments. 13

The piece-wise linear production function employed here is monotonic and concave; it is described fully in GET.

22

Table 4.4: Summary statistics of the σv estimate from the simulation analysis

SFA translog (exponential)

SFA Cobb-Douglas (exponential)

Average of σν across all simulations 0.054 0.108

Standard deviation of σν across all simulations 0.040 0.054

Instances of zero σν 21 0

MAD scores (for reference) 0.061 0.073

MSE scores (for reference) 6.49 9.86

The results show that the translog SFA model, which is the most accurate of

the SFA models under these conditions with regards to productivity change

estimates according to GET, displays an average estimate of σν that is very

close to its true value. However, the standard deviation of this average

measure is quite large; the 95% upper confidence interval is approximately

0.135, which is more than twice as large as the true value. The simulation

analysis also finds that out of the 100 simulation experiments, in 21 of those

the translog SFA models displayed an estimated σν that was approximately

equal to zero. This suggests that sometimes even the more accurate SFA

model is not able to detect the presence of noise, even though modest levels

of noise are part of the DGP. For the Cobb-Douglass SFA model, there were

no instances where σν approached zero, but the σν estimate was also twice as

large on average as the true standard deviation of the noise component.

Overall, the results from the simulations demonstrate that in conditions that

approximate those found in the current analysis, the estimate of σν can

provide an overall indication of the extend of measurement error/noise in the

data, with the caveat that high levels of precision should not be expected.

4.4 Are the parametric models miss-specified?

Table 4.5 below provides the results of the RESET test and the p-values of

the coefficients from the parametric models.

23

Table 4.5: Statistical significance of the variables in the parametric models and RESET test results from the application

COLS (Cobb-Douglas)

COLS (translog)

SFA (Cobb-Douglas, half-normal)

SFA (Cobb-Douglas, exponential)

SFA (translog, half-normal)

SFA (translog, exponential)

L 0.00 0.00 0.00 0.00 0.00 0.00

K 0.00 0.00 0.00 0.00 0.00 0.00

t 0.00 0.17* 0.00 0.00 0.65* 0.71*

L2 0.00 0.00 0.00

K2 0.00 0.00 0.00

t2 0.30* 0.32* 0.33*

LK 0.00 0.00 0.00

Kt 0.01 0.00 0.00

Lt 0.04 0.00 0.00

Insignificant variables 0 2 0 0 2 2

RESET p>F 0.00 0.00

The analysis found that both the Cobb-Douglas and the translog models failed

to pass the RESET test; in addition, all translog models found that the time

variable and its square displayed coefficients that were statistically

insignificant. Both of these factors suggest that the parametric models could

suffer from some form of miss-specification.

The next step is to test whether parametric models that are known to be miss-

specified also display similar symptoms; this is achieved by a round of

simulation experiments that use the same assumptions as those in section

4.3. The following table provides a summary of the instances of statistically

insignificant variables and failed RESET tests from the simulations.

24

Table 4.6: Summary of statistical significance of the variables in the parametric models of the simulation analysis

COLS (Cobb-Douglas)

COLS (translog)

SFA (Cobb-Douglas, exponential)

SFA (translog, exponential)

L 0 3 0 0

K 0 1 0 1

t 79 96 59 68

L2 40 20

K2 27 12

t2 95 71

LK 17 8

Kt 94 68

Lt 91 65

Average number of insignificant variables 0.79 4.64 0.59 3.13

Cases where all variables were significant 21 0 41 22

Cases where RESET failed 40 51 N/A N/A

The simulation analysis shows that the RESET test found evidence of miss-

specification in almost half of the simulation experiments. In addition, there

were instances of insignificant variables in the majority of the experiments

undertaken; the translog COLS specification had no cases where all variable

were significant, while the Cobb-Douglas SFA model that (correctly) assumed

exponentially-distributed inefficiency was the better performing model in this

measure, with just 41 cases where all variables were statistically significant.

Overall, these results suggest that when the parametric models suffer from

functional form miss-specification, possible symptoms include statistical

insignificant variables and failures in the RESET test. Given that similar

symptoms where observed in the current application, one could conclude that

the parametric models in this application are likely to suffer from some form of

miss-specification, which would negatively impact the accuracy of their

productivity change estimates.

4.5 Selecting the most appropriate estimation approach

With regards to the characteristics of the current dataset, this analysis found

that:

25

– input volatility is quite low, averaging just 1.7% p.a. for the labour input

and 0.9% p.a. for the capital input (section 4.1);

– average technical inefficiency across all approaches in this application is

approximately 82% (section 4.2);

– the SFA models suggest that the standard deviation of the normally-

distributed noise component (σν) probably takes a value between 0.05

and 0.1 (section 4.3);

– the parametric models are likely to suffer from some form of miss-

specification, which could be due to the adopted functional form not being

an appropriate representation of the underlying DGP section 4.4);

According to the above findings, the simulation experiment from GET that

more closely matches the characteristics of the current dataset is S2.3 with

‘default’ input volatility. In more detail, for the S2.3 simulation experiment:

– the underlying DGP is piecewise-linear, since the current analysis found

that neither the Cobb-Douglas nor the more flexible translog functional

forms provide a close approximation to the underlying DGP.

– inputs are scaled from one year to the next by a random factors that

follows N~(0,0.1), which results in input volatility similar the EU KLEMS

dataset.

– average technical efficiency in the simulations is designed to be

approximately 87% on average - the current analysis found that average

technical efficiency across all approaches in the EU KLEMS dataset is

82%.

– includes a noise component in the DGP, which is randomly generated

and follows N~(0,0.05). The decision to adopt this level of noise could be

considered conservative, since the mid-point between the various chosen

estimates of σν is closer to 0.075.

The summary accuracy measures of the above experiment are replicated in

the table below:

26

Table 4.7: Summary accuracy results

Measure GA COLS COLS (translog) DEA SFA

SFA (translog)

SFA (half-normal)

MAD 5.80% 6.30% 6.50% 5.80% 7.10% 6.10% 6.40% Accuracy scores MSE 5.33 6.24 7.81 5.23 9.11 6.12 6.96

MAD 1 5 5 1 7 3 5 Accuracy rankings MSE 2 4 6 1 7 3 4

Note: MAD = Mean Absolute deviation, MSE = Mean Square Error

As table 4.7 demonstrates, the two most accurate approaches in this

simulation experiment were DEA and GA, closely followed by the translog

SFA model. The DEA and GA accuracy scores are almost identical; it should

be noted however that the simulation analysis is designed such that the

relevant input and output prices indices required by GA are measured with no

error, while also explicitly assuming that there is no element of allocative

inefficiency in the analysis. The reason for designing the experiment in such a

way was that it allowed for a level playing field when comparing the GA with

the frontier-based estimates, which do not rely on price information. In a real

life application such as the current analysis however, some amount of

measurement error is expected to be present in the price data; in addition, the

GA estimates would also be influenced by changes in allocative inefficiency in

the countries examined. Given that the impact of those factors to the relative

accuracy of the GA estimates under the current conditions is unknown, it

would be more prudent to rely mostly on the DEA-based productivity

estimates.

5 Summary and conclusions

The aim of this paper was to devise a selection framework to help policy

makers choose the productivity measurement approach that is likely to

produce the most accurate estimates relative to the application in hand. This

selection framework includes three steps:

– First, determine those conditions/characteristics inherent in the DGP that

can have a significant influence in the relative accuracy of the assessed

productivity measurement approaches.

– Secondly, examine the current dataset and try to quantify said

conditions/characteristics.

27

– Finally, examine the relative accuracy of the adopted approaches in

datasets specifically designed to display those characteristics/factors

found the real-life data of the current application.

With regards to the fist step, we rely on the findings of the simulation analysis

undertaken in GET, which identified that the characteristics of the DGP that

are most influential the overall accuracy of the most common productivity

measurement approaches include: input volatility, technical inefficiency, noise

and whether the parametric approaches are likely to suffer from functional

form misspecification. The above list is not necessarily exhaustive and here

may well be additional characteristics that have a significant impact on the

overall accuracy of productivity change estimates, such as latent

heterogeneity amongst the assessed units and variable returns to scale. We

leave the assessment of these and other potentially significant characteristics

for future research.

At the second step, we attempt to assess whether and to what extent the

above characteristics are present in the application at hand. To do so we

propose the use of a number of well-established diagnostics and indicators so

that the proposed selection framework can be easily implementable, such as

RESET for assessing functional form miss-specification and estimates of

technical efficiency derived from the assessed approaches. As was

mentioned before, the results of such analysis should not be taken as

absolutes, especially regarding the noise estimates and the presence of

functional form miss-specification. Nevertheless, the simulation analysis

undertaken in this paper does demonstrate that such can provide relatively

reliable estimates. Hopefully, more focused diagnostics/indicators can be

developed and refined in the future.

The third and final step of the selection framework is to determine which of the

assessed approaches is more accurate overall, under the conditions prevalent

in the application in hand. In this paper, this was achieved by relying on the

findings of GET; however, if the application at hand is quite dissimilar to the

various DGP adopted in past simulation studies, our recommendation would

be to construct a new DGP that more closely matches with the current

conditions and use that as the basis of a new round of simulations.

28

As an example of how this selection framework can be implemented in

practice, we assess the productivity performance of a number of EU countries

using the EU KLEMS dataset. The analysis found that although at first glance

all assessed approaches (namely Growth Accounting, Circular DEA-based

Malmquist indices and COLS- and SFA-based Malmquist indices) produce

very similar productivity change estimates on average, the productivity

estimates from the various approaches are quite dissimilar at the individual

country level. These differences are problematic from a policy perspective,

since policy decisions on the issue of economic growth rely on having

accurate productivity estimates at the national level. By applying the proposed

selection framework, it was possible to derive that the approach that is likely

to provide the most accurate estimates in this instance was the DEA-based

Malmquist indices. We propose that such a selection framework (hopefully

refined and expanded in future iterations) can help improve our understanding

of the complex issue of productivity change.

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